Chapter 4: Propositional Proofs

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Chapter 4 Propositional Proofs

• Based on Harry Genslers book
• For CS2209, 2009
• By Dr. Charles Ling cling_at_csd.uwo.ca

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4.1/4.2 Easier Proofs

• Formal proofs main way to prove arguments
• Use rules to syntactically derive valid wffs
• But using I/S-rules in Chapter 3 is incomplete
• A new way prove by contradiction
• To prove a conclusion, assume the opposite
• Then try to derive a contradiction
• RAA, reductio ad absurdum (reduction to
  absurdity).
• Then we prove the conclusion!
• Prove the number of prime numbers is infinite

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• 1. The only people in the mansion were the butler
  and the maid.
• 2. If the only people in the mansion were the
  butler and the maid, then the butler or the maid
  did it.
• 3. If the maid did it, then she had a motive.
• 4. The maid didnt have a motive.
• Prove butler committed the murder
• Assume that the butler didnt do it.
• 5. Assume The butler didnt do it.
• 6. ? Either the butler or the maid did
  it.     from 1 and 2
• 7. ? The maid did it.     from 5 and 6
• 8. ? The maid had a motive,     from 3 and 7
• This leads to a contradiction Thus the butler
  did it.

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What Rules to Use?
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Key components in a proof

• A premise is a line consisting of a wff by itself
  (with no asm or ?).
• An assumption is a line consisting of asm and
  then a wff.
• A derived step is a line consisting of ? and
  then a wff.
• A formal proof is a vertical sequence of zero or
  more premises followed by one or more assumptions
  or derived steps, where each derived step follows
  from previously not-blocked-off lines by RAA or
  one of the inference rules listed above, and each
  assumption is blocked off using RAA.
• Two wffs are contradictory if they are exactly
  alike except that one starts with an additional
  .

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4.2a Exercise
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4.2b Exercise

• 1. If we had an absolute proof of Gods
  existence, then our will would be irresistibly
  attracted to do right.
• If our will were irresistibly attracted to do
  right, then wed have no free will.
• ? If we have free will, then we have no
  absolute proof of Gods existence.  
• (Use P, I, and F)
• 2. If the world had a beginning in time and it
  didnt just pop into existence without any cause,
  then the world was caused by God.
• If the world was caused by God, then there is
  a God.
• There is no God.
• ? Either the world had no beginning in time,
  or it just popped into existence without any
  cause.  (Use B, P, C, and G)

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4.3 Easier refutations

• If we try to prove an invalid argument, we wont
  succeed instead, well be led to refute the
  argument by finding a set of assignment making
  the premises all true and conclusion false

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Strategy for simple proof

• 1 START Assume the opposite of the conclusion.
• 2 SI Derive whatever you can using the S- and
  I-rules, until you get nothing more
• 3 RAA If you get a contradiction, apply RAA and
  you have proved the original conclusion.
• 4. REFUTE Else construct a refutation box.

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4.3a Exercise
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4.3b Exercise

• 1. If the butler shot Jones, then he knew how to
  use a gun.
• If the butler was a former marine, then he
  knew how to use a gun.
• The butler was a former marine.
• ? The butler shot Jones.     Use S, K, and
  M.
• 2. If predestination is true, then God causes us
  to sin.
• If God causes us to sin and yet damns sinners
  to eternal punishment, then God isnt good.
• ? If God is good, then either predestination
  isnt true or else God doesnt damn sinners to
  eternal punishment. Use P, C, D, and G

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4.4 Multiple assumptions

• We may still get stuck Solution make
  another assumption

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• ASSUME Make another assumption if you have an
  unstarred, notblocked-off wff of one of these
  forms for which you dont already have one side
  or its negation
• (AB) (A?B) (A?B)
• Assume one side or its negation
• asm A asm A asm B asm B
• Then return to step 2 (SI).

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This proof strategy can prove or refute any
propositional argument.

• It is sound and complete
• 1 START Assume the opposite of the conclusion.
• 2 SI Derive whatever you can using the S- and
  I-rules. If you get a contradiction, Step 3
  otherwise Step 4 otherwise Step 5
• 3 Apply RAA and prove the original conclusion
• 4. ASSUME Make another assumption, Step 2
• 5. REFUTE If you dont get a contradiction,
  construct a refutation box.

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An Example
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4.5a Exercise
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4.5b Exercise

• 1. If President Nixon knew about the massive
  Watergate cover-up, then he lied to the American
  people on national television and he should
  resign.
• If President Nixon didnt know about the
  massive Watergate cover-up, then he was
  incompetently ignorant and he should resign.
• ? Nixon should resign. Use K, L, R, and
  I.
• 2. The parents told their son that their
  precondition for financing his graduate education
  was that he leave his girlfriend Suzy. A friend
  of mine talked the parents out of their demand by
  using this argument.
• If you make this demand on your son and he
  leaves Suzy, then hell regret being forced to
  leave her and hell always resent you.
• If you make this demand on your son and he
  doesnt leave Suzy, then hell regret not going
  to graduate school and hell always resent you.
• ? If you make this demand on your son, then
  hell always resent you.      Use D, L, F, A,
  and G

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4.6 Harder refutations

• Multiple-assumption invalid arguments

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Other Proof Systems

• Many sound and complete proof systems for prop
  logic
• Traditional Proofs More rules